Vol. 303, No. 2, 2019

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Invariant connections and PBW theorem for Lie groupoid pairs

Camille Laurent-Gengoux and Yannick Voglaire

Vol. 303 (2019), No. 2, 605–667
Abstract

Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e., a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space LA. For Lie groupoid pairs with vanishing Atiyah class, we show that the left A-action on the quotient space LA can be linearized.

In addition to giving an alternative proof of a result of Calaque about the Poincaré–Birkhoff–Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to simultaneous linearization of all the monodromies.

We also develop a general theory of connections on Lie groupoid equivariant principal bundles.

Keywords
Atiyah classes, Lie groupoids, homogeneous spaces, linearization, Poincaré–Birkhoff–Witt theorem, foliations, equivariant principal bundles
Mathematical Subject Classification 2010
Primary: 53C05, 53C30
Secondary: 53C12
Milestones
Received: 28 March 2018
Revised: 4 June 2019
Accepted: 4 June 2019
Published: 4 January 2020
Authors
Camille Laurent-Gengoux
Institut Élie Cartan de Lorraine (UMR 7502)
Université de Lorraine
Metz
France
Yannick Voglaire
Mathematics Research Unit
Université du Luxembourg
Maison du Nombre
Esch-sur-Alzette
Luxembourg